Weekly Problems 2018-B

Initially only 1…Qxg8 and 1…Qe8 are without set mates, and the key 1.Be7! (waiting) completes the block by crossing over f6 so that a knight check on that square wouldn’t cause a self-interference. The black queen’s focus on e5 and f6 is lost in most cases when it moves, e.g. 1…Qh6/Qf7/Qe5/Qxc4 (creating a flight on c6) 2.Se5 and 1…Qxe2/Qd5/Qf6/Qxg8 (creating a flight on e8) 2.Sf6. When the focus is maintained by the queen, new mates result: 1…Qxf5 2.Qxf5, 1…Qxe7 2.dxe7, but 1…Qxd6 allows not only the intended 2.Qxd6 but also 2.Se5/Sf6 due to the pin. The black bishop induces three more mates: 1…Bb6 2.Sxb6, 1…Bc7 2.Rxc7, and 1…Bd8 2.Rxd8. Except for the 1…Qxd6 variation, the play is completely dual-free.

Andy Sag: The key is the only move that completes the block of eight set mates by preventing 1…Qe8 and leaving Black with 17 legal moves. Note the symmetry between ten of the queen moves and knight mates.
Jacob Hoover: The anticritical key 1.Be7! crossing the critical square f6 makes no threat but puts Black in zugzwang. All the set mates are preserved.
George Meldrum: After scanning the board from left to right, we finally come to the key piece. It then moves from the h-file to the e-file, backwards so to speak.
Nigel Nettheim: The key completes the block by ruling out 1...Qe8, so it is perhaps not especially subtle; but the variations are nice.
Ian Shanahan: Here we see the focal theme exhibited by the black queen in action. The element of symmetry detracts somewhat.

Paz Einat and I independently came up with this correction, which eliminates the triple mate after 1…Qxd6 by making it a check, forcing 2.Qxd6. The original bishop key is replaced by 1.Qd3! Two further advantages of this version are that it saves two pawns and adds three plausible tries – 1.Qf4? Qxe7!, 1.Qf3? Qxd6+!, and 1.Qe5? Qxc4! – a mini duel between the queens.

Every black move in the diagram has a set mate except for 1…Sxe7. The square-vacating key 1.Bb8! threatens 2.Sc7. Since the threat-move cuts off the bishop, any move by the d6-knight would defend by creating a potential flight. The random move 1…Sd~ permits 2.c4. Two correction moves by the piece show subtle dual avoidance. First, 1…Sc4 is a self-block that enables White to self-interfere with the h4-rook, but 2.Sdf4? gives a flight on e4, while 2.Sef4 compensates by opening the e7-rook’s line to the same square. Second, 1…Se4 again self-blocks and now 2.Sef4? fails due to the flight on c4, but 2.Sdf4 works by opening the f1-bishop’s line to that square. The sophisticated line-play rendered – involving white mating moves that simultaneously open and close white lines while exploiting black self-blocks – is known as Theme E. There’s one minor variation, 1…Sd4 2.Rxd4.

Andy Sag: The sacrificial key vacates c7 to threaten 2.Sc7 and thus provide for 1…Sxe7. The main feature seems to be the two self-blocks allowing alternate knights to mate on f4 using indirect batteries to guard the unblocked squares.
Nigel Nettheim: A neat scheme.
Ian Shanahan: A complex tangle of line-openings and -closings involving “valves” and the Russian line-play themes popular in the 1930s. A really fine problem in Meredith!

Regardless of where is the missing king, White has no mate in two if black castling is available as a defence. The placement of the white king is thus aimed at preventing this black move, though not directly, but through retrograde analysis. Add the king on a7, and since a problem position is assumed to have arisen from the normal starting array, we can ask: how did the king get to a7? The black pawns on b7 and c7, which have never moved, preclude a route via a6 or b6. So to reach a7 the white king must have travelled through the 8th rank via d7 or d8, in either case displacing the black king from its starting square. That means the black king has moved previously, and castling is now illegal. The key 1.Qd5! grants two flights on e7 and f8 – or three if castling were legal – and threatens 2.Qxf7, which remains playable after the flight-moves. Two variations follow: 1…Sg5 2.Qd8 and 1…Rf8 2.Qd7.

Andy Sag: White king positions on the 8th rank [a8, b8, and c8] also disable 1…0-0, but are ruled out due to the defence 1.Qd5? Ke7+!
Jacob Hoover: 1…Sg5 interferes with the h4-bishop and 1…Rf8 self-blocks.
George Meldrum: A great fun problem placing the white king. Numerous tries for the two-mover [1.Qxc7? Be7!], and more when you place the white king on c8 [1.Rd7? Rf8!].
Nigel Nettheim: A nice way to present a simple retro idea.
Dennis Hale: An elegant elimination of castling.
Ian Shanahan: A simple yet cute retro!

Four set variations account for all possible black moves in the diagram: 1…Kc6 2.Qb7, 1…Se~ 2.Qe4, 1…c6 2.Qd4, and 1…Sg~ 2.Bf3, so this is a complete block. White has no way of preserving all of these lines though, e.g. 1.Bd1? Se2!, 1.Be3? c5! The key 1.Qb5! (waiting) removes the flight on c6 but creates a new one on e4: 1…Ke4 2.Qc4. The e7-knight now exhibits correction play, 1…Se~ 2.Re5 and 1…Sc6 2.Bb4, bringing about two changed mates. A third change occurs with 1…c6 2.Qd3, while 1…Sg~ 2.Bf3 is as set. An especially difficult mutate with some surprising changes.

Andy Sag: The give-and-take key changes three mates as well as the flight. Noted the white pawn on a5 prevents the 1.Sa5 (2.Qc4) cook.
Jacob Hoover: The c5-bishop has numerous tries (1.Bf2/Be3/Bxa7?), all of which produce the changed mates 1…Se~ 2.Qc5 and 1…Kc6 2 Qc5, but they're all defeated by 1…c5!
Nigel Nettheim: The refutation of the try 1.Bd1? is neat. After 1.Qb5! Ke4 the pin of the e7-knight is irrelevant, perhaps a slight weakness.
Ian Shanahan: A beautiful mutate with a key that gives and takes a flight. Lovely (though the position is rather crowded, as is so often the case with Stocchi).

If the two f-pawns were removed in the diagram position, the f8-rook could mate, and indeed it’s not possible to arrange any other mating configurations, given the fixed positions of the white units. Black must capture the f6-pawn with the king, since to do so with the rook would give check (forbidden during the sequence). This plan requires the black king to be shielded from all three white line-pieces: 1.Rg2 2.Kg1 3.Kf1 4.Re2 5.Re3 6.Ke2 7.Kd3 8.Kd4 9.Re7 10.Rf7 11.Kd5 12.Ke6 13.Kxf6. Now if Black captures the f4-pawn too soon, the opened f-file would mean the king cannot return to h1, as the black rook is unable to shield the king from both white rooks simultaneously. So the king goes back first: 14.Ke6 15.Kd5 16.Kd4 17.Re7 18.Re3 19.Kd3 20.Ke2 21.Kf1 22.Re2 23.Rg2 24.Kg1 25.Kh1. Finally the f4-pawn can be captured, followed by the rook’s return to h2: 26.Rf2 27.Rxf4 28.Rf2 29.Rh2 for 29…Rf1. An impressive black minimal problem that features numerous shields and rundlaufs by both black pieces.

Andy Sag: The mate is not hard to see, but how to get rid of the f-pawns in as few moves as possible avoiding checks leads to a unique series.
George Meldrum: A tortuous path but one without compromise on move order or move direction.
Ian Shanahan: A beautiful and perfectly constructed example of well-known ideas: king-shields, encirclement, a long trek with subtle manoeuvring and precise switchbacks.

After the key 1.d6!, Black can stop the threat of 2.Qxe5 by capturing the knight on d3. Such a capture would self-block and apparently enable the white rook to give various battery mates by moving along the f-file. But each capture by a black piece involves an additional effect that obliges the rook to play to one specific square: 1…Bxd3 2.Rf1, 1…Sxd3 2.Rf2, 1…cxd3+ 2.Rf7, and 1…Rxd3 2.Rf4. The multiple self-blocks on the same square lead to differentiated white mates or dual avoidance – a combination known as the Stocchi theme. There is by-play with 1…Rd5 2.Qe3.

Andy Sag: The key allows a check. This and three other set self-blocks after capture of the d3-knight require four different positions of the white rook when firing the battery. Note the rook capture requires a double check. Finally, the rook blocks the threat but again opens the line to e3 as well as creating a self-block on d5.
Jacob Hoover: If we were to replace the d3-knight with a dummy black piece, White would have five different mates: 2.Rf8, Rf7[A], Rf6, Rf2[B], Rf1[C]. Three of the four captures of the knight force a particular one of these rook discoveries: [A], [B] and [C]; thus, we have an incomplete Stocchi block since there is no capture that forces 2.Rf8 or 2.Rf6. The fourth capture, however, forces a new rook discovery: 1…Rxd3 2.Rf4.
Nigel Nettheim: Not hard to solve, because the a2-g8 diagonal was likely to come into play. But the four captures on d3 are very nicely handled.
George Meldrum: A quiet unassuming key move. Four defensive captures on d3, all with unique mating responses. Very nice.
Ian Shanahan: Four-fold Stocchi blocks – self-block plus dual avoidance – on d3. From memory, the record is five. An excellent problem!

This miniature is solved by 1.Ra5 Bc1 2.Ka4 Kc4 3.Rb3 axb3 (ideal-mate) and 1.Rab3 axb3 2.Ka3 Kc3 3.Ra2 Bc5. The two phases show exchange of functions performed by both white and black pieces. The white bishop and pawn take turns to guard a flight and to give mate, while the two black rooks swap the tasks of sacrificing on b3 and blocking a flight.

Andy Sag: A role reversal study.
Jacob Hoover: In each solution each unit moves exactly once and the kings move in the same direction.
George Meldrum: Mate with the bishop seemed most likely. Stumbled over the pawn mate solution but found the bishop mate solution most elusive. Both mates are very neat indeed.
Ian Shanahan: White’s non-royal units exchange functions between the solutions (known as a funktionwechsel). A fine and pretty miniature!

The astonishing key 1.Ke2! not only permits the black pawn to promote with check, but surprisingly also entails no threats! Neither 2.Kf2? nor 2.Kf1? is threatened because of 2…d1=S+! and 2…d1=Q+! respectively. Rather, White waits for the pawn to commit to a particular promotee, which then rules out the alternative. After 1…d1=Q+ 2.Kf2 (threat: 3.Rh6), the queen has eight different checks, all answered with precise captures – 2…Qe2+ 3.Sxe2, 2…Qxf3+ 3.Sxf3, 2…Qd2+ 3.Sxd2, 2…Qc2+ 3.Sxc2, 2…Qxd4+ 3.Sxd4, 2…Qe1+ 2.Sxe1, 2…Qf1+ 3.Kxf1, and 2…Qg1+ 3.Sxg1 – followed by 3…g4 4.Rh6. Against 1…d1=S, White doesn’t continue with 2.Kf1? Se3+!, but 2.Rxg5 (3.Rg1) Sc3+ 3.Kf1 Se2 4.Rh6. The weakness of 1…g4 is distant self-block: 2.Rh6+ Kg2 3.Rh2+ Kg3 4.Sf5. Lastly 1…Kg2 leads to 2.Rxg5+ Kh3 3.Be6 or 2…Kh1 3.Rg1. The underutilised white bishop is the only flaw in this very tough four-mover, reminiscent of Sam Loyd’s three-move classic, “Steinitz’s Gambit.”

Andy Sag: I tried all other moves first and no joy. The key 1.Ke2!!! allows promotion and if to a queen, the next move leaves no safe checks and the queen is powerless to stop the mate.

After the key 1.Rf4!, the threat of 2.Qe4 induces Black to defend with the two half-pinned pieces on the long diagonal: 1…Sc5 2.Sc7, 1…Sd6 2.Sxb6, 1…Re6 2.Qb5, and 1…Rc4/Rxc8 2.Qd7. White exploits not only Black’s self-pins in these thematic variations but also the self-blocks (by the knight) and self-block/unguards (by the rook). The black bishop triggers the by-play, 1…Be5 2.Qxe5 and 1…Bd4 2.Rxd4.

Andy Sag: The half-pin sets up two symmetrical pairs of pin-mates.
Jacob Hoover: The rook variations are changed from the set 1…R~file/Re6 2.Qe6. The set dual 1…Sd6 2.Sxb6/Qe6 is also removed by the key.
Nigel Nettheim: Easy to solve, because the rook had to remain on its rank. The half-pin on the long diagonal is well exploited.
Ian Shanahan: An economical rendering of a typical Good Companions idea: complete half-pin with three thematic self-blocks. And a fine, retreating key thrown in for good measure.

In this complete block position, set mates are prepared for all black moves: 1…Sb~ 2.S1c2, 1…Sxd3 2.Qxd3, 1…Sh~ 2.Sf3, and 1…fxe1=Q 2.Qf4. But White cannot maintain all of the set play with any waiting move, and the key 1.Qh1! involves a threat, 2.Qe4. Two defences by the b-knight bring about changed play, due to the different squares controlled by the queen: 1…Sc2+ 2.S3xc2 and 1…Sxd3 2.S1c2. One further line is as set: 1…Sg2/Sf3 2.Sf3. A good example of the block-threat type.

Andy Sag: A complete block with (arguably) three changed mates, the third one being the threat after 1…fxe1=Q (set 2.Qf4).
Jacob Hoover: 1…Sxd3 2.S1c2 is also a transferred mate (relative to the set 1…Sb~ 2.S1c2).
Nigel Nettheim: Neat changed-mates for the b-knight. Not hard to solve, because the h1-a8 diagonal was inviting.
George Meldrum: The white queen abandons its defence of d3 which initially looked essential to respond to the black knight moving to c2 and d3. New enjoyable variations now cover those moves with all other lines being supplemental.
Ian Shanahan: A fine block-threat (all mates set, but the key threatens) with two changed mates. The key took longer to find than I expected.

The diagram position is solved by 1.Rb4 Kd2 2.Kc4 Bf7. When the h5-bishop is replaced by a knight, the solution becomes 1.Ke4 Sc6 2.Rd5 Sg3. Both parts finish with an ideal-mate, delivered by the starting piece on h5. Only the e3-bishop is a static piece in this miniature with an open setting.

Andy Sag: Not hard when you realise that at least two of the white pieces don’t move thus limiting the number of final scenarios to look at.
Jacob Hoover: In part (a) the black rook has to move across the critical square c4 before the king moves there, and in part (b) the king must vacate d5 so that the rook can move there.
George Meldrum: Neat.
Ian Shanahan: An attractive miniature showing two different ideal mates. Sweet!

The diagram is a complete block, and the only move that doesn’t disrupt any of the set play is 1.Rd2! (waiting). A random move by the black bishop unguards d4: 1…B~ 2.Rd4. The first correction 1…Be3 self-blocks, permitting 2.Sg3. The second correction 1…Bc5 is an anticipatory interference with the c6-rook, enabling the white queen to unpin the piece directly with 2.Qb1. This effect is known as a Gamage unpin, and it’s seen again in 1…Sd6 2.Qxe7, and a third time in 1…e6 2.Qh7. Another self-block occurs with 1…e5 2.Sg5, while 1…Sb5 2.Qxc6 makes further use of the white queen. A well-constructed waiter with no extra materials needed to control the white king or the f8-rook, both of which execute tries defeated by one of the thematic defences: 1.Kb3? Bc5! and 1.Rf7? e6!

George Meldrum: Good array of mates in the set play. Simple to solve.
Jacob Hoover: 1…Bc5 and 1…e6 open white lines and close black ones.
Ian Shanahan: The key piece has to tread carefully, after which the main thematic content is the three direct unpins by the white queen after Black's anticipatory interferences. I also enjoyed the two “secondary corrections” by the black bishop. An excellent problem in the Good Companions style.
Andy Sag: If you shift the f2-pawn to g5 and add a white pawn on e3, we have an eighth variation, 1…g4 2.Rf4. Now you get a pair of short-range lateral rook mates (following unguards) to complement the pair of knight mates (following self-blocks) and four queen mates from different directions.

The farsighted key 1.Kg2! (waiting) opens the first rank for eventual access by the queen while avoiding a prospective interference with the same piece (1.Kf2?). After 1…Ka1, 2.Qd4 (waiting) pins the pawn and 2…Kb1 results in a short mate, 3.Qd1 (hence not 2.Qc3? Kb1!). The main variation continues with 2…Bb1 and now 3.Qg1! pins the bishop and puts Black in zugzwang yet again, forcing 3…Ka2 4.Qa7. Attractive sweeping play by the white queen, making the most of the piece’s power. Note also how the two black pieces exchange their positions.

Andy Sag: The key allows the queen to operate unobstructed. Almost a one-liner. A neat miniature four-mover.
George Meldrum: A super first move by White with the king making way for the queen to eventually use the first rank.
Jacob Hoover: The final position is a model mate [and so is the short variation mate].
Bob Meadley: A beauty and definitely a classic. Not a check to be seen until the end.

After the key 1.Se2!, White threatens 2.Sxf4. Random moves by the f4-bishop lead to dual mates, which are separated by 1…Be3 2.Rxe3 and 1…Bc1 2.Sxc1. Correction play by the bishop disables these mates but admits new ones, twice by unpinning the white queen: 1…Bg5 2.Qxc3 and 1…Be5 2.Qe3, and once by blocking a flight: 1…Bd2 2.Bc2. The black knights have three defences, and two of them also unpin the queen: 1…Se5 2.Qd4, 1…Sd5 2.Qc2, and 1…Sxe2 2.Bxe2. A fifth unpin, via direct withdrawal by the pinning piece, occurs with 1…Qc4/Qxe8 2.Qc4. The black queen produces some by-play as well: 1…Qxc5+ 2.Sxc5 and 1…Qb4 2.Sxb4. A pity about the dual, 1…Qb8 2.Qc4/Qxc3 (duplicating queen mates already seen), which can be removed by adding a black pawn on b7.

Andy Sag: The threat plus eleven variations make this a busy problem. The h6-pawn preventing 1…Bh6 gives a clue to the solver. Better to have a black pawn on c6 instead of the a4-bishop for stopping the 1…Sd5 2.Qc2/Qxb5 dual, and then the f7-pawn can be removed [as 1…Qxe8+ is ruled out].
Jacob Hoover: The dual after 1…Qb8 is unfortunate, but the very rich thematic content more than makes up for that. I wonder if that dual can be removed?
George Meldrum: Not a great deal of new mates after the key yet still satisfying in its complexity. Had hoped that 1.Sh1 had been the key.
Nigel Nettheim: Wonderful! So many satisfying ingredients!
Ian Shanahan: Various unpins of the white queen combined with black correction by the menaced bishop. A lovely strategic problem!

The key 1.Qb3! entails a sacrificial threat, 2.Qe6+ fxe6 3.fxe6. The c6-rook and e2-bishop defend on c4, causing a mutual interference between the two pieces. After 1…Rc4, 2.Qxb5 threatens 3.Qe5/Qe8, both stopped by 2…Rc5, but the rook has opened a line for the white queen and also cut off the b6-bishop, allowing 3.Qxe2, a capture of the other thematic black unit (or 2…Bd4/Rd5 3.Qd5). We find a beautifully matching variation in 1…Bc4 2.Qxc2+ Bd3 – opening a line for the queen and cutting off the d1-rook – enabling 3.Qxc6, which captures the thematic rook (or 2…Rd3 3.Sd2). The by-play consists of some short variations, 1…Rc5 2.Qe3, 1…Bd4/Rd5 2.Qd5, and 1…Bxf3 2.Qxf3.

Andy Sag: Try 1.Qa8? (threat: 2.Qe8) Bxf3! A well disguised threat involving a queen sacrifice. Tricky play after defensive blocks on c4. Three short mates detract slightly.
Jacob Hoover: Black has only two defenses, and they constitute a Grimshaw pair.
George Meldrum: Solving this problem is like taking a mouthful of sherbet powder and having your cheeks explode. The setting has more rough edges than Mount Everest; however, the jewels in the crown are after Black plays either the rook or bishop to c4 where play is amazingly done. I like it.

The set play 1…S~ 2.Bc6 is abandoned by the key, 1.Sc6! which grants a flight on d7 and threatens 2.Qe7. Taking the flight with 1…Kd7 prompts 2.Qe6. The black knight produces two variations: 1…Sg6 2.Qf7 – a change from the set – and 1…Sxc6 2.Bxc6, bringing back the set mate. And 1…Bd6/Bd8 permits 2.Qd8. Including the threat, there are four “close-encounter” queen mates.

Andy Sag: Miniature with flight-giving key and five mates (including threat).
Jacob Hoover: 1…Bd8/Bd6 allows 2.Qd8, an echo mate to the threat. 1…Sxc6 2 Bxc6 is a model mate.
Nigel Nettheim: Flight-giving and neat. Only six pieces are needed, but the white king placement adds tries.
Ian Shanahan: A fine, flight-giving key leading to four pretty variations. Sweet indeed! The American Bob Lincoln was probably the most prolific miniaturist ever.

Initially the c4-rook and b6-bishop are preventing knight mates on f4 and f2 respectively. Three tries and the key cut off one of these black pieces to threaten mate: 1.Be4? (2.Sf4) Be3!, 1.Bc5? (2.Sf2) Bxc5+!, 1.Sd4? (2.Sf2) Rd5!, and 1.Sc5! (2.Sf2). The key changes the response for 1…Rf4+ from the set 2.Sxf4 to a battery mate, 2.gxf4. The remaining defences by the two thematic black pieces are gate-openings for the white queen: 1…Bxc5 2.Qh6 and 1…Rxc3/Re4 2.Qf1.

Andy Sag: The key blocks the black bishop’s guard on f2 and the black rook’s line to d5, opens the 6th rank and changes the mate for the set check.
Ian Shanahan: The key changes the set mate to 1…Rf4+, closes some black lines, and allows the ambushed white queen to be activated by line-openings. Rather heavy, but a fine problem!
Bob Meadley: Frank was 82 at the time.

The knights are preventing respective queen mates in the initial position, and if Black is to move, neither knight could maintain control of the mating square: 1…Se~ 2.Qxd5 and 1…Sd~ 2.Qc7. However, with White to play there is no way of preserving both of these set lines. The key 1.Ra5! (waiting) unguards b6 so that after 1…Se~, the set mate no longer works, but because the rook now attacks b5, 2.cxd5 becomes viable. The other variation is unchanged: 1…Sd~ 2.Qc7. About as simple as it gets for a two-move mutate, but delicately done.

Ian Shanahan: A sweet miniature mutate (i.e., complete block, with at least one mate changed by the waiting key). Here, the mate after 1...Se~ is changed.
Andy Sag: A simple but neat miniature waiter. The key maintains the black king’s confinement, forcing one of the knights to move.
Jacob Hoover: White has several waiting moves that preserve the set play – e.g. 1.Rb1?/Rb7? – but 1…Kc5! refutes all of these.
George Meldrum: This one had me going for a bit. The rook moving to protect the b5-square enabling the changed mate is so elegant.
Nigel Nettheim: Easy to solve, perhaps not so easy to compose. The highlight is the changed mate after 1…Se~.
Bob Meadley: Not bad at all. I know this fellow was a keen Australian Problemist member and will try and put a pen picture together.

In the first solution, Black starts with 1.Rf2 to interfere with the g1-bishop’s control of the eventual mating square. The rook move also allows the white queen to follow along the same line with 1…Qe2, an effect called the Loshinsky’s magnet. The black queen then self-blocks a flight and the white queen mates: 2.Qg4 Qe3. The second solution displays analogous strategy. 1.Bf7 cuts off the f8-rook and clears a line for the white queen, which follows with 1…Qe6. The black queen self-blocks again but on a different square, enabling another queen mate: 2.Qg3 Qf5. Together the two phases demonstrate an orthogonal-diagonal transformation.

Andy Sag: In each case, black piece interference unguards the mating square, white queen follows in Bristol fashion, black queen self-blocks, and white queen mates.
Jacob Hoover: It is interesting to note that in each solution whichever black piece moves first, the white queen moves the same number of squares in the same direction, and in both solutions that number is four!
Nigel Nettheim: A well-matched pair, for each first move shuts off a defender and opens an attacking path.
George Meldrum: Classic interference themes will never fail to please.
Ian Shanahan: Two strategically matched solutions, making an “ortho-diagonal transformation” (ODT). A lovely unified problem with echoed strategies.

Five prominent set variations make use of the royal battery: 1…Rxd4+ 2.Kxd4, 1…Bxc5 2.Kxc5, 1…Qxb5+ 2.Kxb5, 1…Se~ 2.Kd3, and 1…Rxc3+ 2.Kxc3. The key 1.Kb3! unpins the d4-pawn and threatens 2.d5. Two defences prompt changes from the set play: 1…Rxd4 2.Sxd4 and 1…Bxc5 2.Sc7, where the new mates are possible because the key has also unpinned the b5-knight. The royal battery fires three times from the king’s new position: 1…Qxa3+ 2.Kxa3, 1…Qa4+ 2.Kxa4, and 1…Rxc3+ 2.Kxc3. There’s plenty of by-play: 1…Rd1 2.Qc4, 1…f4 2.Qh3, 1…Re5 2.Rxe5, and 1…Sf6 2.Sg7. This classy work could even be improved by shifting the c1-rook to c2, where it would produce another battery variation, 1…Rxb2+ 2.Kxb2.

Andy Sag: The unpinning key sets up (arguably) four changed mates, including the threat in the cases of 1…Qxb5 (set 2.Kxb5) and 1…Se~ (set 2.Kd3).
Jacob Hoover: The key, while not too obvious, was easy to find for me, as I noticed that a couple of black moves in the diagram that didn't have set mates prepared could be answered if the queen and b5-knight weren't pinned.
Nigel Nettheim: The white king is immobilising three of his own units, so he graciously steps back to free them. The g8-knight is not needed, but provides an extra variation.
Ian Shanahan: Mates set for the royal battery are changed by the white king walking along the battery line. A cornucopia of variations, from one of the most prolific two-move composers ever! A lovely problem.

After the key 1.Qg8!, the threat of 2.Qxh7 can only be stopped by the black knights, which are half-pinned by the white bishop. When either knight moves, the other is fully pinned, a weakness that White exploits in 1…Sd4 2.Sc3 and 1…Se5 2.Sf6, besides the self-blocking errors. One further defence by the c6-knight is a simple unguard: 1…Sce7 2.Qe6. Any move of the other knight defends by opening the rank for the a5-rook, and a random placement results in a third pin-mate, 1…Sd~ 2.Qc4. The correction move 1…Se3 self-blocks another flight, permitting 2.Rf4.

Jacob Hoover: Both the key and a thematic try attempt to exploit the half-pinned knights. The try 1.Qh8? threatens 2.Qxh7, but 1…Sf6! refutes it.
Andy Sag: The inactive white queen and the half-pinned knights give strong clues. The e1-bishop and d2-pawn prevent a dual, 1…Sd4 2.Sc3/Sg3, but a black pawn on h4 will do the same job.
George Meldrum: Pretty obvious that the queen is the key piece with a choice of three or four candidate squares. Very nice knightly play with four new mates introduced. The black bishop on a4 is unneeded as 1.Bxc6 can be answered with 1…Bg3.
Nigel Nettheim: The a4-bishop and the pawns on a7 and c7 are not needed. The long diagonal has been filled, which looks nice and may have been taken as a task.
Ian Shanahan: A classy half-pinner with a rich strategic blend in the Good Companions style 40 years on.

In part (a), Black begins with a self-block, 1.Rf4, and White unpins the black knight with 1…Sd4, a move that also guards two flights. The unpinned knight plays 2.Sd5, both to interfere with the a2-bishop and to allow the white bishop to protect the knight on d4, and White mates with the rook, 2…Re6. The twin (b) has the white knight starting on h2 instead, and the resulting solution involves analogous strategic effects. Now Black self-blocks with 1.Be6, and then 1…Rd4 uses the rook to unpin the black knight and control four flights. The freed knight cuts off the black queen with 2.Se2 and again enables the white bishop to control d4, after which the white knight mates with 2…Sg4.

Andy Sag: In (b) the white rook and knight reverse roles.
Jacob Hoover: The white rook and knight swap roles (guarding/mating) between the two parts.
George Meldrum: High esteem for this problem as both parts finish with model mates.

The key 1.Se5! discards a set variation, 1…bxc4 2.Rxc4, and threatens 2.Rd3. 1…Qe3 blocks a potential flight and enables 2.Sb3, a battery mate that interferes with a3-rook’s control of e3. 1…Bxe4 is another self-block that allows the battery to fire again: 2.Sf3, which cuts off the g2-bishop’s guard of e4. A third thematic variation, 1…Sc5 2.Sc6, sees a self-block on c5 followed by a mating move that closes the c8-rook’s line to the same square. The excellent by-play shows a pair of knight promotions causing an interference between Black’s queen and rook, which are protecting each other against the white queen: 1…e1=S 2.Qxa1 and 1…c1=S 2.Qxg1.

Andy Sag: Two sub-promotion defences where first-rank interference allows a pair of diagonal queen mates and two self-blocks on the e-file allow a pair of double-check mates. The fifth defence self-block allows the key piece to mate. The g3-pawn appears to be unnecessary.
Jacob Hoover: Three defenses self-block, and two of the resulting mates are battery mates.
Ian Shanahan: Three variations all involve self-blocks plus white interferences. Two more parade knight promotions that interfere with black line-pieces symmetrically. A fine problem from a great stalwart of the Good Companions.

Set mates are provided for all possible black moves in the diagram. In particular, moves by the c1-bishop allow 2.Sd2, though 1…Bxe3 gives a dual, 2.Sd2/Qxe3. White has no way of maintaining all of the set play, however, and the sacrificial key 1.Qb1! (waiting) generates some changes besides adding the variation, 1…cxb1=Q 2.Bd1. Now a random bishop move results in a dual, 1…B~ 2.Sd2/Qxh1 and these mates are separated by 1…Bd2+ 2.Sxd2 and 1…Bxe3 2.Qxh1. The black knights’ variations are unchanged from the set: 1…Sb7+ 2.Qxb7, 1…Sc6+ 2.Bxc6, 1…Sf7 2.Qb7/Bc6, and 1…Sh~ 2.Rf2. The brilliant key compensates for the duals (in both the set and actual play), which are a flaw especially in a mutate as here.

Andy Sag: The sacrificial key entices line clearance for the a4-bishop.
Ian Shanahan: An astonishing, quiet, sacrificial key. Technically, the problem is a mutate (with a single change after 1…Bxe3, though the set play after this defence appears to be dualled, sadly).

Part (a) is solved by 1.Gf1 Gd1 2.Ge1 Gd4. The black grasshoppers self-block on f1 and e1, with the second move enabled by white grasshopper acting as a hurdle on d1. Hence the try 1.Gf1 Gb4? – aiming for 2…Gh4 – fails because the c1-grasshopper cannot reach e1. For part (b), shift the white grasshopper to d1, and the solution becomes 1.Ge1 Ga4 2.Gf1 Gh4. The twinning here is paradoxical because even though in (a) the white grasshopper moves to d1 in the solution, when the piece begins on d1 in (b), that original solution no longer works. And the only reason it fails is that White lacks a tempo move, i.e. the try here is 1.Ge1 White tempo?? 2.Gf1 Gd4.

Paz Einat: Nice exchange of black move order. In the first solution Black must wait for White to make the hurdle for the c1-grasshoper, while in the second the same piece on c1 must jump first as the white grasshopper will move away.
George Meldrum: A fantastic and pleasing introduction to grasshoppers for someone who has not solved such problems (like me). All moves are by grasshoppers!

Black’s pieces require a minimum of four moves to reach their places in the diagram, while White apparently needs only one move to get the knight to c3, since the missing f-pawn can be captured on its initial square. White must use up the three spare moves somehow, but simply shifting a piece back and forth wouldn’t work as that will take up an even number of moves. The only way to waste three tempi exactly is to move the f-pawn to another square where it can also be captured. 1.f4 e5 2.f5 Bc5 3.f6 Qxf6 4.Sc3 Qf2. Tempo play is a popular idea in proof games but the specific theme shown here – which I called a tempo sacrifice – is rarely seen.

Andy Sag: Scholar’s Mate by Black in four moves.
Jacob Hoover: The pawn’s little trek forces the move order quite nicely.
Bob Meadley: Oh woe is White. Nice.
George Meldrum: Easy and elegant, end of year problem.
Ian Shanahan: Sweet! I'm no expert on SPGs, but I found this to be really pretty. Solving it was a nice rush! The white f-pawn gains tempi and is captured.